## April 16, 2020

#### Posted by David Corfield

We began a discussion of Jacob Lurie’s Ultracategories over here, in particular whether they may be construed as algebras for some 2-monad. Perhaps this topic deserves a post to itself, rather than appearing tucked at the end of a long and fascinating discussion about condensed/pyknotic mathematics.

I have just discovered a 1995 PhD thesis by Francisco Marmolejo, advised by Robert Paré, that’s very relevant. Unfortunately the only online access is to a very poor photocopy, here. Anyway, Marmolejo characterises Makkai’s ultracategories there in a 2-monadic way. This would still leave Lurie’s somewhat differently defined ultracategories on the to-do list.

There’s then a follow-up question of characterising any such 2-monads using the codensity monad construction if possible. There’s some ongoing codensity conversation over here.

This area isn’t just for categorical logicians – some philosophers are paying attention, see

• Halvorson, Hans and Tsementzis, Dimitris (2015), Categories of scientific theories, (preprint)

especially pp. 20-21.

…there seems to be nothing inherently “category-theoretic” about ultraproducts. As Makkai’s work proves and Los’ theorem has long made obvious, taking ultraproducts is a fundamental operation when it comes to elementary classes: elementary classes are exactly those classes closed under elementary equivalence and the taking of ultraproducts. Since every pretopos corresponds to an elementary class (more precisely: to the category of models of a coherent theory) one would imagine that any such characterization of Pretop would amount to a characterization of “closure under ultraproducts”. Absent any useful purely categorical description of ultraproducts (or even ultrafilters) this seems like a significant obstruction. Nevertheless the work of Leinster (2013) on ultrafilter monads as codensity monads might provide a way out, though this is still very far from being made precise.

Finally, I still have an open question about how to characterise the $\infty$-version developed by Lurie as a logical result about syntax-semantic duality. What kind of ‘theory’ is involved?

Posted at April 16, 2020 11:55 AM UTC

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Section 4.5.3 of Marmolejo’s thesis tells us of a 2-adjunction between $PRETOP^{op}$ and $CAT$, with 2-functors $Mod(-)$ and $Set^{(-)}$. Given an algebra for the resulting 2-monad, $T$, on $CAT$, that is, a category, $A$, and functor $\Phi: Mod(Set^A) \to A$, then given an ultrafilter $(I, U)$, we can form an ultraproduct in $A$ via the ultraproduct map, $Mod(Set^A)^I \to Mod(Set^A)$.

This gives us a pre-ultracategory. Then a further 2-adjunction between $PRETOP^{op}$ and $T-ALG$ is used to generate a 2-monad whose algebras give Makkai ultracategories.

Posted by: David Corfield on April 16, 2020 7:19 PM | Permalink | Reply to this

Indications from Mike here that ultracategories are a kind of generalized multicategory.

Posted by: David Corfield on July 8, 2020 10:00 AM | Permalink | Reply to this

Taking our cue from Di Liberti’s Codensity: Isbell duality, pro-objects, compactness and accessibility

The codensity monad of the Yoneda embedding is isomorphic to the monad induced by the Isbell adjunction,

I think we should look to the codensity 2-monad of the Yoneda lemma for bicategories. Kan extensions for bicategories are explained in Enriched categories as a free cocompletion, sec. 10.

Then varying the choice of 2-category in the Yoneda embedding, varying the class of limits preserved, etc. as in the ordinary case of Isbell duality should generate many 2-adjunctions along with their 2-adjoint dualities, including Gabriel-Ulmer duality and Marmolejo’s above.

James Dolan and John Baez seemed to be on this track in some notes on Doctrines:

There is a finite limits theory $Fin\Set$, which is the embodiment of propositional logic. All finite sets can be formed from $2$ by products and equalizers, so this finite limits theory is the Cauchy completion of the Lawvere theory whose objects are powers of $2$, which is the opposite of the category of finitely presented Boolean algebras. Note: $[Fin\Set, Set] = \Bool\Alg$ is the opposite of the category of profinite sets. $[Fin\Set^{op}, Set] = Set$ since $Fin\Set^{op}$ is the free finite limits theory on one object.

Categorifying this example, we should get an interesting doctrine which is the embodiment of predicate logic. Namely, let $\FP\Gpd$, the $(2,1)$-category of finitely presented groupoids. Claim: theories of this doctrine are theories of first-order predicate logic. Conjecture: $[\FP\Gpd, Gpd]$ is the opposite of the category of profinite groupoids. $[\FP\Gpd^{op}, Gpd] = Gpd$. With luck this will explain the appearance of profinite groups in number theory.

James and John work with $Gpd$ rather than $Cat$, but assuming things work out, we’d have finite limit preserving 2-functors $[\FP\Cat^{op}, Cat] = Cat$ sitting in a 2-adjunction with $[\FP\Cat, Cat]^{op}$, presumably with $Set$ acting as dualizing object, as we see with Marmolejo. Other choices would include $[\mathbf{1}, Cat] = Cat$ in duality with its opposite.

As Di Liberti shows us, pro-finite objects and compactness are often involved in one side of an Isbell duality. Then the remark about the appearance of profinite groups in number theory is intriguing. Is it that the interest in condensed/pyknotic structures arises from the nature of one side of categorified Isbell duality?

Posted by: David Corfield on April 18, 2020 9:18 AM | Permalink | Reply to this

To remind myself of what $Cat^{op}$ is, Mike sketched an answer back here:

“cocomplete categories equipped with a strongly-generating set of tiny objects, and functors having both left and right adjoints (equivalently, by the adjoint functor theorem, preserving small limits and colimits) whose left adjoint preserves the chosen generators.”

Posted by: David Corfield on April 22, 2020 8:17 AM | Permalink | Reply to this

Interesting, there’s a theory of pro-2-objects.

• M. Emilia Descotte, Eduardo Dubuc, A theory of 2-pro-objects (with expanded proofs), (arXiv:1406.5762)

I’ve just started an nLab page for it.

So what would $2Pro(FP Cat)$ look like?

The construction hasn’t been taken up much so far, but there is

• Yuliang Huang, Giulio Orecchia, Matthieu Romagny, Unramified F-divided objects and the étale fundamental pro-groupoid in positive characteristic, (arXiv:1906.05072)

who use 2-pro-objects to construct the étale fundamental pro-groupoid of a flat finitely presented algebraic stack.

Posted by: David Corfield on April 22, 2020 10:27 AM | Permalink | Reply to this

Continuing to plough this furrow, I’ve come to believe that 2-Isbell duality is simply formal, as suggested at the end of ‘Proof B’ on the nLab page Isbell duality, using the 2-Yoneda embedding. So for a small 2-category, $\mathcal{A}$, there is Isbell duality between $Cat^{\mathcal{A}^{op}}$ and $(Cat^{\mathcal{A}})^{op}$.

Then we should be able to imitate Di Liberti’s section 2.2 and consider cases of $\mathcal{A}$ as a dense sub-2-category of $\mathcal{K}$ to produce the corresponding codensity 2-monad via the composition of the 2-adjunction between $\mathcal{K}$ and $Cat^{\mathcal{A}^{op}}$ and the Isbell 2-adjunction. If all that works, what follows if $FP Cat$ is a dense sub-2-category of $Cat$?

The $Set$ case has the advantage that finite and finitely-presented coincide there, leading to the results in sec 2.3 of Di Liberti’s paper.

But what then follows from $FinSet$ as finite objects of $Set$? That’s the topic of Sec 3 on compactness:

all the algebras for the codensity monads of finite structures admit … a compact Hausdorff structure and that this happens coherently with the natural compact Hausdorff structure that pro-objects have.

Perhaps I should also look at $FinCat$.

Posted by: David Corfield on April 26, 2020 12:40 PM | Permalink | Reply to this

One of the instances listed in Isbell duality is that between commutative rings and affine schemes. If there is a 2-Isbell duality to be had as above, one would expect a categorification of this duality to feature. Naturally something along these lines has already appeared, see 2-algebraic geometry.

One interesting paper there is

There we find commutative 2-rings defined in duality with affine 2-schemes, such that any Grothendieck topos is a commutative 2-ring.

In Sec 6.2 the authors consider $C \mapsto Hom_{Com2Ring}({}^{C}Set,Set)$, for a small category $C$, where ${}^{C}Set$ is $Fun[C, Set]$. As above $Set$ is playing its role as a dualizing object. We are told this 2-functor “will be the topic of future work”. Was it, I wonder?

Posted by: David Corfield on April 22, 2020 2:23 PM | Permalink | Reply to this

Urs had already identified such a 2-duality as a categorified Isbell duality back here, in more detail at nLab: Tannaka duality for geometric stacks. Note, this is in terms of a $(2,1)$-functor. See also sec 6.2 of Chirvasitu and Johnson-Freyd.
Proposition 3.5.4 of Tensor functors between categories of quasi-coherent sheaves, the dual adjunction between $2-Ring$ and $Stack$ as “entirely formal and an example of (higher) Isbell duality”.